Empirical saddlepoint approximations of the Studentized ratio and regression estimates for finite populations

We obtain saddlepoint approximations for tail probabilities of the Studentized ratio and regression estimates of the population mean for a simple random sample taken without replacement from a finite population. This is only possible if we know the entire population, so we also obtain empirical saddlepoint approximations based on the sample alone. These empirical approximations can be used for tests of significance and confidence intervals for the population mean. We compare the empirical approximation to the true saddlepoint approximation, both theoretically and numerically. The empirical saddlepoint is related to a bootstrap method for finite populations and we give numerical comparisons of these. We show that for data which contains extreme outliers, poor approximations can be obtained in the case of regression estimates, both for the saddlepoint and empirical saddlepoint, but for less extreme data the saddlepoint and empirical saddlepoint approximations are extremely close to the corresponding Monte Carlo and bootstrap approximations.